Symmetry, invariants, topology. Basic tools

Elementary concepts of group actions: orbits and their stabilizers, orbit t ypes and their strata are introduced and illustrated by simple examples. We give the unified description of these notions which are often used in the different domains of physics under different names. We also explain some ba sic facts about rings of invariant functions and their module structure. Th is leads to a geometrical study of the orbit space and of the level surface s of invariant functions (e.g. energy levels of Hamiltonians). Combining th ese tools with Morse theory we study the extrema of invariant functions. So me physical applications (not studied in other chapters) are sketched. (C) 2001 Elsevier Science B.V. Ail lights reserved.